Is the universe computable?

From: David Barrett-Lennard <dbl.domain.name.hidden>
Date: Tue, 4 Nov 2003 11:45:39 +0800

In the words of Tegmark, let's assume that the physical world is
completely mathematical; and everything that exists mathematically
exists physically.
 
I have been thinking along these lines since my days at university -
where it occurred to me that any alternative is mystical. However, the
problem remains to explain induction - ie the predictability of the
universe. Why is it that the laws of physics can be depended on when
looking into the future, if we are merely a "mathematical construction"
- like a simulation running on a computer. It seems to me that in the
ensemble of all possible computer simulations (with no limits on the
complexity of the "laws") the ones that remain well behaved after any
given time step in the simulation have measure zero.
 
Given the "source code" for the simulation of our universe, it would
seem to be possible to add some extra instructions that test for a
certain condition to be met in order to tamper with the simulation. It
would seem likely that there will exist simulations that match our own
up to a certain point in time, but then diverge. Eg it is possible for
a simulation to have a rule that an object will suddenly manifest itself
at a particular time and place. The simulated conscious beings in such
a universe would be surprised to find that induction fails at the moment
the simulation diverges.
 
In other words, at each time step in a simulation the state vector can
take different paths according to slightly different software programs
with special cases that only trigger at that moment in time. It seems
that a universe will continually split into vast numbers of child
universes, in a manner reminiscent of the MWI. However there is a
crucial difference - most of these spin off universes will have bizarre
things happen. It is difficult to see how a computable system can be
tamper proof. How can a past which has been well behaved prevent
strange things from happening in the future?
 
In the thread "a possible paradox", there was talk about a vanishingly
small number of "magical" universes where strange things happen.
However, it seems to me that the bigger risk is that a "normal" universe
like ours will be the atypical in the ensemble!
 
A possible argument is to invoke the anthropic principle - and suggest
that our universe is predictable in order for SAS's to evolve and
perceive that predictability. However, that predictability only needs
to be a trick - played on the inhabitants for long enough to develop
intelligence. There is no reason why the trick needs to continue to be
played.
 
I suggest that the requirement of a tamper-proof physics is an extremely
powerful principle. For example, we deduce that SAS's only exist in
mathematical systems that aren't computable. In particular our
Universe is not computable.
 
- which is what Penrose has been saying.
 
I have assumed that non-computability coincides with being tamper-proof
but this is far from clear. For example, it is conceivable that the
Universe is a Turing machine running an infinite computation (cf
Tipler's Omega point), and "awareness" only emerges in the totality of
this infinite computation. Perhaps our awareness is a manifestation of
advanced waves sent backwards in time from the Omega Point!
 
I think it's important to distinguish between an underlying mathematical
system, and the formal system that tries to describe it. I think this
is a crucial distinction. For example, the real number system can be
defined uniquely by a finite set of axioms. Uniqueness is (formally)
provable - in the sense that it can be shown that an isomorphism exists
between all systems that satisfy the axioms. However the real numbers
are uncountably infinite - and therefore are very poorly understood
using formal mathematics - which is limited to only a countably infinite
set of statements about them. So formal mathematics should be regarded
as an imperfect and coarse tool which only gives us limited
understanding of a complicated beast! This is after all what Godel's
incompleteness theorem tells us.
 
It is not surprising that a computer will never exhibit awareness -
because it is merely using the techniques of formal mathematics, and not
tapping into the "good stuff".
 
- David
 
 
 
 
Received on Mon Nov 03 2003 - 22:47:33 PST